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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 6.1 Introduction to the Normal Curve HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Normal Distribution: A continuous probability distribution for a given random variable, X, that is completely defined by it’s mean and standard deviation. Properties of a Normal Distribution: 1. A normal curve is symmetric and bell-shaped. 2. A normal curve is completely defined by its mean, , and standard deviation, . 3. The total area under a normal curve equals 1. 4. The x-axis is a horizontal asymptote for a normal curve. HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Symmetric and Bell-Shaped: HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Completely Defined by its Mean and Standard Deviation: An inflection point is a point on the curve where the curvature of the line changes. The inflection points are located at - and + . HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Total Area Under the Curve = 1: HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve The x-Axis is a Horizontal Asymptote: HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Determine if the following is a normal distribution: a. Birth weights of 75 babies. Normal b. Ages of 250 students in 10th grade. No, this would be uniform c. Heights of 100 adult males. Normal d. Frequency of outcomes from rolling a die. No, because the data is discrete e. Weights of 50 fully grown tigers. Normal HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve How Many Normal Curves are there? Because there are an infinite number of possibilities for and , there are an infinite number of normal curves. HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Standard Normal Distribution: A standard normal distribution has the same properties as the normal distribution; in addition, it has a mean of 0 and a standard deviation of 1. Properties of a Standard Normal Distribution: 1. The standard normal curve is symmetric and bellshaped. 2. It is completely defined by its mean and standard deviation, = 0 and = 1. 3. The total area under a standard normal curve equals 1. 4. The x-axis is a horizontal asymptote for a standard normal curve. HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Converting to the Standard Normal Curve: Standard Score Formula (z-score): When calculating the z-score, round your answers to two decimal places. HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Draw a Normal Curve: Given = 40 and = 5, indicate the mean, each of the inflections points, and where each given value of x will appear on the curve. x1 = 33 and x2 = 51 Solution: 35 33 40 45 51 HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Convert to the Standard Normal Curve: Given = 40 and = 5, calculate the standard score for each x value and indicate where each would appear on the standard normal curve. x1 = 33 and x2 = 51 Solution: -1 -1.4 0 1 2.2 HAWKES LEARNING SYSTEMS Continuous Random Variables math courseware specialists 6.1 Introduction to the Normal Curve Convert to the Standard Normal Curve: Given = 48 and = 5, convert to a normal curve and indicate where a score of x = 45 would appear on each standard normal curve. Solution: 43 45 48 53 -1 -0.6 0 1